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Definition/Summary
Feynman diagrams are a combinatorial device for enumerating a sum of integrals of products of delta functions.
Feynman diagrams are a mathematical tool with no physical significance. In particular, no single Feynman diagram is intended to represent any physical interaction: the physical interaction is represented, to a required order of accuracy, by the sum of the values of all the Feynman diagrams of that order.
These delta functions occur in calculating the coefficients of the Dyson expansion of the S-matrix element for a transition between two free-particle states.
They result from moving each annihilation operator through many other annihilation and creation operators to the far right of an expression, where it will ultimately annihilate the vacuum state, resulting obviously in a contribution of zero. On the way there, every time one operator moves past another, a term must be added with both operators omitted and replaced by a delta function multiplied by a "propagator" value.
Calculation of this propagator value is generally made easier by a mathematical trick involving "off-mass-shell" 4-momentums, but such momentums have no physical significance.
The process continues until nothing is left but a sum of products of delta functions and propagator values.
It is a sum because there is a different product for each order in which the operators can be moved through each other.
Each Feynman diagram corresponds to a different order of movement: the Feynman diagrams are a way of keeping track of these different orders, and of assigning the correct products of propagator values to each one.
A Feynman diagram may be labelled in two ways: in the position representation, each vertex is labelled with a space-time position, and the whole diagram is integrated over those positions, while in the momentum representation, each external line is labelled with an "on-mass-shell" 4-momentum, and each internal line is labelled with an "off-mass-shell" 4-momentum, and the whole diagram is integrated over those "off-mass-shell" 4-momentums.
Equations
Extended explanation
Dyson expansion:
This is intended as an outline only, not as a rigorous explanation.
The Dyson expansion gives the S-operator in terms of a sum of time-ordered integrals of products of the potential:
[tex]S\ =\ \sum_{N\ =\ 1}^{\infty} (-i)^N\int_{-\infty}^{\infty}\cdots\int_{-\infty}^{t_3}\int_{-\infty}^{t_2}V(t_1)\cdots V(t_N)\,dt_1\cdots dt_N[/tex]
These integrals over time can be rewritten as ordinary integrals of time-ordered products of the Hamiltonian density over space and time:
[tex]S\ =\ \sum_{N\ =\ 1}^{\infty}\frac{(-i)^N}{N!}\ \int_{-\infty}^{\infty}\cdots\int_{-\infty}^{\infty}T\{H(x_1)\cdots H(x_N)\}\,d^4x_1\cdots d^4x_N[/tex]
The symbol [itex]T\{\cdots\}[/itex] denotes time-ordering, meaning that the order of the Hamiltonian densities is changed so that the [itex]x_i[/itex]s are in order of decreasing magnitude of their time-components: in other words, so that those later in time are on the left.
It can be proved that this time-ordering is Lorentz invariant for non-commuting Hamiltonian densities, and for commuting Hamiltonian densities it clearly does not make any difference anyway.
For a transition between two free-particle states, we must insert the inner product containing the vacuum state [itex]\Phi_0[/itex] and the annihilation operators for the initial state and the creation operators for the final state:
[tex]S_{\mathbf{p}',\mathbf{p}}\ =\ \sum_{N\ =\ 1}^{\infty}\frac{(-i)^N}{N!}\ \int_{-\infty}^{\infty}\cdots\int_{-\infty}^{\infty}\left(\Phi_0,\ \prod a(\mathbf{p}'_i)\,T\{H(x_1)\cdots H(x_n)\}\prod a^\dagger (\mathbf{p}_j)\,\Phi_0\right)\ d^4x_1\cdots d^4x_N[/tex]
In this expression, the annihilation operators are all on the left, and the creation operators on the right. There are also annihilation and creation operators in the field operators inside the Hamiltonian densities. We must now use commutation or anti-commutation relations to move each of the annihilation operators to the far right, where it will annihilate the vacuum state, resulting obviously in a contribution of zero. This leaves only a sum of products of delta functions multiplied by propagators.
Propagators:
To save space, particle operators above have been indexed only by 3-momentum. In this section, they will be more fully indexed by 3-momentum [tex]\bold{p}[/itex] spin-component [tex]\sigma[/itex] and species [tex]n[/itex]
The simplest propagator is between the "bare" operators on the far left and far right:
[tex]a(\mathbf{p}',\sigma ',n')a^\dagger(\mathbf{p},\sigma ,n)\ =\ \mp a^\dagger(\mathbf{p},\sigma ,n)a(\mathbf{p}',\sigma ',n')\ \ +\ \ \delta^3(\mathbf{p}'\ -\ \mathbf{p})\delta_{\sigma '\sigma}\delta_{n'n}[/tex]
Propagators between a "bare" operator and a field, or between two fields, involve integrals in their calculation, and the results are listed in "Equations" above.
Choosing the diagram:
Label each vertex with a type of interaction (if there is more than one type in the Hamiltonian density).
Choose the number of vertices of each type equal to the required order in each type of interaction.
Draw one external line for each initial or final particle, marked with an arrow pointing in or out respectively.
Join each external line to one vertex, or to another external line of the opposite type (ie initial to final), and also join the vertices (in pairs) by internal lines, each marked with an arrow, so that each vertex has the number of lines appropriate to that type of interaction.
For example, for the contribution of the third order in one type and the fourth order in another, to the interaction between two particles, draw diagrams with four external lines and with three vertices labelled "1" and four vertices labelled "2", each with the number of lines appropriate to that type of interaction.
Labelling by vertex (position representation):
Label each vertex with a space-time position (for integrating over).
Label each end of each line where it meets a vertex with a field type (so an internal line may have two different field labels), and with the field's value at that space-time position.
Assign a coupling factor [itex]-ig[/itex] to each vertex, where g is the appropriate coupling constant. Assign a propagator factor to each line.
Multiply all the factors for one diagram, and integrate over all possible space-time positions.
Labelling by internal line (4-momentum representation):
Label each internal line with an "off-mass-shell" 4-momentum (for integrating over).
Assign a coupling factor [itex]-ig(2\pi)^4\,\delta^4(\sum q)[/itex] to each vertex, where the sum inside the delta function is of the 4-momentums of all the lines at that vertex, each multiplied by ±1 according to whether its arrow points in or out, and whether it represents a particle or anti-particle.
Assign a matrix [tex]-i(2\pi)^{-4}\,P_{lk}(q)/(q^2\,+\,m^2\ -\ i\epsilon)[/tex] to each internal line with arrow pointing from end-type k to end-type l.
And to each external line with arrow pointing from end-type k or to end-type l, assign a factor [itex](2\pi)^{-3/2}[/itex] times the "vector" appropriate for that field-type and for that initial or final particle (such as [itex]u_l(\mathbf{p},\sigma ,n)[/itex]).
For scalars, [itex]P(q)\ =\ I[/itex] and the "vectors" are [itex](2q^0)^{-1/2}[/itex]. For spinors, [itex]P_{lk}(q)\ =\ (-i\gamma_{\mu}p^{\mu}\,+\,m)\beta_{lk}[/itex] and the "vectors" are normalised Dirac spinors.
Multiply all the factors for one diagram, integrate over all possible 4-momentums, and sum over all field-types.
* This entry is from our old Library feature. If you know who wrote it, please let us know so we can attribute a writer. Thanks!
Feynman diagrams are a combinatorial device for enumerating a sum of integrals of products of delta functions.
Feynman diagrams are a mathematical tool with no physical significance. In particular, no single Feynman diagram is intended to represent any physical interaction: the physical interaction is represented, to a required order of accuracy, by the sum of the values of all the Feynman diagrams of that order.
These delta functions occur in calculating the coefficients of the Dyson expansion of the S-matrix element for a transition between two free-particle states.
They result from moving each annihilation operator through many other annihilation and creation operators to the far right of an expression, where it will ultimately annihilate the vacuum state, resulting obviously in a contribution of zero. On the way there, every time one operator moves past another, a term must be added with both operators omitted and replaced by a delta function multiplied by a "propagator" value.
Calculation of this propagator value is generally made easier by a mathematical trick involving "off-mass-shell" 4-momentums, but such momentums have no physical significance.
The process continues until nothing is left but a sum of products of delta functions and propagator values.
It is a sum because there is a different product for each order in which the operators can be moved through each other.
Each Feynman diagram corresponds to a different order of movement: the Feynman diagrams are a way of keeping track of these different orders, and of assigning the correct products of propagator values to each one.
A Feynman diagram may be labelled in two ways: in the position representation, each vertex is labelled with a space-time position, and the whole diagram is integrated over those positions, while in the momentum representation, each external line is labelled with an "on-mass-shell" 4-momentum, and each internal line is labelled with an "off-mass-shell" 4-momentum, and the whole diagram is integrated over those "off-mass-shell" 4-momentums.
Equations
Extended explanation
Dyson expansion:
This is intended as an outline only, not as a rigorous explanation.
The Dyson expansion gives the S-operator in terms of a sum of time-ordered integrals of products of the potential:
[tex]S\ =\ \sum_{N\ =\ 1}^{\infty} (-i)^N\int_{-\infty}^{\infty}\cdots\int_{-\infty}^{t_3}\int_{-\infty}^{t_2}V(t_1)\cdots V(t_N)\,dt_1\cdots dt_N[/tex]
These integrals over time can be rewritten as ordinary integrals of time-ordered products of the Hamiltonian density over space and time:
[tex]S\ =\ \sum_{N\ =\ 1}^{\infty}\frac{(-i)^N}{N!}\ \int_{-\infty}^{\infty}\cdots\int_{-\infty}^{\infty}T\{H(x_1)\cdots H(x_N)\}\,d^4x_1\cdots d^4x_N[/tex]
The symbol [itex]T\{\cdots\}[/itex] denotes time-ordering, meaning that the order of the Hamiltonian densities is changed so that the [itex]x_i[/itex]s are in order of decreasing magnitude of their time-components: in other words, so that those later in time are on the left.
It can be proved that this time-ordering is Lorentz invariant for non-commuting Hamiltonian densities, and for commuting Hamiltonian densities it clearly does not make any difference anyway.
For a transition between two free-particle states, we must insert the inner product containing the vacuum state [itex]\Phi_0[/itex] and the annihilation operators for the initial state and the creation operators for the final state:
[tex]S_{\mathbf{p}',\mathbf{p}}\ =\ \sum_{N\ =\ 1}^{\infty}\frac{(-i)^N}{N!}\ \int_{-\infty}^{\infty}\cdots\int_{-\infty}^{\infty}\left(\Phi_0,\ \prod a(\mathbf{p}'_i)\,T\{H(x_1)\cdots H(x_n)\}\prod a^\dagger (\mathbf{p}_j)\,\Phi_0\right)\ d^4x_1\cdots d^4x_N[/tex]
In this expression, the annihilation operators are all on the left, and the creation operators on the right. There are also annihilation and creation operators in the field operators inside the Hamiltonian densities. We must now use commutation or anti-commutation relations to move each of the annihilation operators to the far right, where it will annihilate the vacuum state, resulting obviously in a contribution of zero. This leaves only a sum of products of delta functions multiplied by propagators.
Propagators:
To save space, particle operators above have been indexed only by 3-momentum. In this section, they will be more fully indexed by 3-momentum [tex]\bold{p}[/itex] spin-component [tex]\sigma[/itex] and species [tex]n[/itex]
The simplest propagator is between the "bare" operators on the far left and far right:
[tex]a(\mathbf{p}',\sigma ',n')a^\dagger(\mathbf{p},\sigma ,n)\ =\ \mp a^\dagger(\mathbf{p},\sigma ,n)a(\mathbf{p}',\sigma ',n')\ \ +\ \ \delta^3(\mathbf{p}'\ -\ \mathbf{p})\delta_{\sigma '\sigma}\delta_{n'n}[/tex]
Propagators between a "bare" operator and a field, or between two fields, involve integrals in their calculation, and the results are listed in "Equations" above.
Choosing the diagram:
Label each vertex with a type of interaction (if there is more than one type in the Hamiltonian density).
Choose the number of vertices of each type equal to the required order in each type of interaction.
Draw one external line for each initial or final particle, marked with an arrow pointing in or out respectively.
Join each external line to one vertex, or to another external line of the opposite type (ie initial to final), and also join the vertices (in pairs) by internal lines, each marked with an arrow, so that each vertex has the number of lines appropriate to that type of interaction.
For example, for the contribution of the third order in one type and the fourth order in another, to the interaction between two particles, draw diagrams with four external lines and with three vertices labelled "1" and four vertices labelled "2", each with the number of lines appropriate to that type of interaction.
Labelling by vertex (position representation):
Label each vertex with a space-time position (for integrating over).
Label each end of each line where it meets a vertex with a field type (so an internal line may have two different field labels), and with the field's value at that space-time position.
Assign a coupling factor [itex]-ig[/itex] to each vertex, where g is the appropriate coupling constant. Assign a propagator factor to each line.
Multiply all the factors for one diagram, and integrate over all possible space-time positions.
Labelling by internal line (4-momentum representation):
Label each internal line with an "off-mass-shell" 4-momentum (for integrating over).
Assign a coupling factor [itex]-ig(2\pi)^4\,\delta^4(\sum q)[/itex] to each vertex, where the sum inside the delta function is of the 4-momentums of all the lines at that vertex, each multiplied by ±1 according to whether its arrow points in or out, and whether it represents a particle or anti-particle.
Assign a matrix [tex]-i(2\pi)^{-4}\,P_{lk}(q)/(q^2\,+\,m^2\ -\ i\epsilon)[/tex] to each internal line with arrow pointing from end-type k to end-type l.
And to each external line with arrow pointing from end-type k or to end-type l, assign a factor [itex](2\pi)^{-3/2}[/itex] times the "vector" appropriate for that field-type and for that initial or final particle (such as [itex]u_l(\mathbf{p},\sigma ,n)[/itex]).
For scalars, [itex]P(q)\ =\ I[/itex] and the "vectors" are [itex](2q^0)^{-1/2}[/itex]. For spinors, [itex]P_{lk}(q)\ =\ (-i\gamma_{\mu}p^{\mu}\,+\,m)\beta_{lk}[/itex] and the "vectors" are normalised Dirac spinors.
Multiply all the factors for one diagram, integrate over all possible 4-momentums, and sum over all field-types.
* This entry is from our old Library feature. If you know who wrote it, please let us know so we can attribute a writer. Thanks!